MATH Seminar

Title: Clusters and semistable models of hyperelliptic curves
Seminar: Algebra and Number Theory
Speaker: Jeffrey Yelton of Emory University
Contact: David Zureick-Brown, dzureic@emory.edu
Date: 2021-09-21 at 4:00PM
Venue: MSC W301
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Abstract: For every hyperelliptic curve $C$ given by an equation of the form $y^2 = f(x)$ over a discretely-valued field of mixed characteristic $(0, p)$, there exists (after possibly extending the ground field) a model of $C$ which is \emph{semistable} -- that is, a model whose special fiber (i.e. the reduction over the residue field) consists of reduced components and has at worst very mild singularities. When $p$ is not $2$, the structure of such a special fiber is determined entirely by the distances (under the discrete valuation) between the roots of $f$, which we call the \emph{cluster data} associated to $f$. When $p = 2$, however, the cluster data no longer tell the whole story about the components of the special fiber of a semistable model of $C$, and constructing a semistable model becomes much more complicated. I will give an overview of how to construct``nice" semistable models for hyperelliptic curves over residue characteristic not $2$ and then describe recent results (from joint work with Leonardo Fiore) on semistable models in the residue characteristic $2$ situation.